366 XV NORMED ALGEBRAS AND SPECTRAL THEORY have a = ae' (15.8.3(iii)), and on the other hand a = g(e2) = eg(e) = ea, so that ae eAer, and g =fa. It is clear that eAe' c: eA n Ae'; conversely, if y e eA n Ae!, then j; = j>e' and y = ey (15.8.3(iii)), so that y = eye' e eAe'. The image#(I) is a left ideal contained in I', so it is either {0} or I'; likewise, the kernel ^"^O) is a left ideal contained in I, hence is either {0} or I. If ^-'(O) = I, then g(l) = {0}; if g~l(G) = {0}, then we must have g(l) * {0}, hence^(1) = I' and g is bijective. Finally, if/fl = 0, we have/a(e) = ea = 0; but a e eAe', so that ea = a, and consequently a = 0. (ii) The C-algebra eAe is a closed subalgebra of A (15.8.3(iii)). Since we have seen above that every element of EndA(I) is either zero or invertible, it follows that EndA(I) is a (possibly noncommutative),/?^, and hence the same is true for 'eAe. Clearly e is the unit element of eAe, and because A is a normable algebra (15.1.8) it follows from the Gelfand-Mazur theorem that eAe = Ce. (Hi) If I and Y are not isomorphic, we have eAe' = {0} by (i) above, and in particular eer = 0; hence e and e' are orthogonal (15.8.4). The same is true of I and I' (15.8.9), and II' = {0}. If I and Y are isomorphic, and if g is an isomorphism of I onto I7, then every homomorphism of I into I' is of the form g o u, where u is an endomorphism of I. Hence eAe' is a complex vector space of dimension 1, by virtue of (i) and (ii) above. Clearly II7 is a left ideal con- tained in I7; since it contains eAe' ^ {0}, it must be equal to I7. (iv) Since Ix is the image of I under the homomorphism y^-^-yx of I into A, it is a left ideal isomorphic to I/I7, where Y is the kernel of the above homomorphism. But I7 must be equal to either {0} or I, and so y\—>yx is either zero or injective. (15.8.13) Suppose that A is separable. Then: (i) There exists a finite or infinite sequence (Ifc)fceJ of minimal left ideals, no pair of which are isomorphic, such that every minimal left ideal of A is isomorphic (as an A-module) to some \k. (ii) For each index keJ, the closure of the sum of all the minimal left ideals of A which are isomorphic to \k is a self-adjoint two-sided ideal ak. Every minimal left ideal of the Hilbert algebra ak is a minimal left ideal of A, isomor- phic to lk, and the algebra ak contains no closed two-sided ideals other than {0} and ak. (iii) Each of the algebras ak is the Hilbert sum of a (finite or infinite) sequence of minimal left ideals isomorphic to lk. The algebra A is the Hilbert sum of the sequence (ak)ke], and ahak = {0} whenever h^k. Start with a decomposition of A as a Hilbert sum of minimal left ideals l'n (15,8.11). We take It = 1; and define inductively lk+l to be equal to 1^,)n for all